3
The VCG Mechanism [The Vickrey-Clarke-Groves (VCG) Mechanism] In every general mechanism design environment, there is a DSIC mechanism that maximizes the social welfare. In particular the allocation rule is x(b) = argmax ω Σ i b i (ω) (1); and the payment rule is p i (b) = max ω Σ j≠i b j (ω) – Σ j≠i b j (ω*) (2), where ω* = argmax ω Σ i b i (ω) is the outcome chosen in (1).

4
Discussion of the VCG mechanism  DSIC mechanism that optimizes social welfare for any mechanism design problem !  However, sometimes impractical.  How do you find the allocation that maximizes social welfare. If Ω is really large, what do you do? -m items, n bidders, each bidder wants only one item. -m items, n bidders, each bidder is single-minded (only like a particular set of items). -m items, n bidders, each bidder can take any set of items.

5
Discussion of the VCG mechanism  Sometimes Computational intractable.  If you use approximation alg., the mechanism is no longer DSIC.  Serves as a useful benchmark for more practical approaches!

7
 Important in practice -spetrum auctions -allocating take-off and landing slots at airports  Notoriously hard in both theory and practice -In theory, many impossibility results for what can be done with reasonable communication and computation -In practice, badly designed combinatorial auctions with serious consequences

8
Combinatorial Auctions (set-up)  n bidders. For example, Bell, Rogers, Telus and several regional providers.  There is a set M of m non-identical items. For example, a license for broadcasting at a certain frequency in a given region.  An outcome is a n-dimensional vector (S 1, S 2,..., S n ), with S i denoting the set of items allocated to bidder i (her bundle). All S i ’s are disjoint!  There are (n+1) m outcomes!!!

9
Combinatorial Auctions (set-up)  Each bidder could value every different outcome differently, but we simplify it a bit here.  i has a private value v i (S) for each subset S of M. Each bidder needs 2 m numbers to specify her valuation. -v i (Ø) = 0 -v i (S) ≤ v i (T), if S is a subset of T. (free disposal) -Could make other assumptions on the valuation function. Usually simplifies the auction design problem. Talk about it later.  The welfare of an outcome (S 1, S 2,..., S n ) is Σ i v i (S i ).

10
Challenges of Combinatorial Auctions  How do you optimize social welfare in combinatorial auctions?  VCG!  Unfortunately, several impediments to implementing VCG.  Challenge 1 -- Preference elicitation: Is direct-revelation sealed-bid auction a good idea?  No! Each bidder has 2 m numbers to specify. When m=20, means 1 million numbers for every bidder.

11
Indirect Mechanisms  Ascending English Auction.  The one you see in movies!  Many variants, the Japanese variant is easy to argue about.  The auction begins at some opening price, which is publicly displayed and increases at a steady rate. Each bidder either chooses “in” or “out,” and once a bidder drops out it cannot return. The winner is the last bidder in, and the sale price is the price at which the second-to-last bidder dropped out.  Each bidder has a dominant strategy: stay till the price is higher than her value.  Apply revelation principle on this auction, you get Vickrey auction.

12
Indirect Mechanisms  We’ll discuss the auction formats used in practice for the spectrum auctions.  Main question: can indirect mechanism achieve non-trivial welfare guarantees?  A lot of work has been done on this front.  Short answer: depends on the bidders’ valuation functions.  For simple valuations, “yes”; for complex valuations, “no”.

14
Challenges of Combinatorial Auctions  Challenge 3: Even if we can run VCG, it can have bad revenue and incentive properties, despite being DSIC.  Example: 2 bidders and 2 items, A and B. -Bidder 1 only wants both items: v 1 (AB) = 1 and is 0 otherwise. -Bidder 2 wants only item A: v 2 (AB) = v 2 (A) =1 and is 0 otherwise. -VCG gives both items to 1 and charges him 1. -Suppose now there is a third bidder who only wants item B: v 3 (AB) = v 3 (B) = 1 and is 0 otherwise. -VCG gives A to 1 and B to 2, but charges them 0! -Can you see a problem?  Vulnerable to collusion and false-name bidding. Not a problem for Vickrey.

15
Challenges of Combinatorial Auctions  Challenge 4: indirect mechanisms are usually iterative, which offers new opportunities for strategic behavior.  Example: bidders use the low-order digits of their bids to send messages to other bidders. -#378 license, spectrum use rights in Rochester, MN -US West and Macleod are battling for it. -US West retaliate by bidding on many other licenses in which Macleod were the standing high bidder. -Macleod won back all these licenses but had to pay a higher price -US West set all bids to be multiples of 1000 plus 378!

17
Selling Items Separately  Indirect mechanisms. Have relax both DSIC and welfare maximization.  Obvious mechanism to try is to sell the items separately, for each, use some single- item auction.  Main take away is: for substitutes this works quite well (if the auction is designed carefully), but not for complements. -substitutes: v(AB) ≤ v(A) + v(B) -complements: v(AB) > v(A) + v(B)  Welfare maximization is computationally tractable when the items are substitutes and true valuations are known. But it’s still intractable for complements.  In real life the items are a mixture of substitutes and complements. When the problem is “mostly substitutes”, then selling items separately could have good performance.

18
Selling Items Separately  Rookie mistake 1: Run the single-item auctions sequentially, one at a time.  Imagine the items are identical and you have k copies.  DSIC mechanism gives the top k bidders each a copy of the item and charge them the (k+1)-th highest bidder’s bid.  What if you run it sequentially? Say k=2.  If you are the highest bidder will you bid truthfully for the first item?  Everyone will do the same reasoning, in the end the outcome is unpredictable.

19
Selling Items Separately  In March 2000, Switzerland auctioned off 3 blocks of spectrum via a sequence of Vickrey auctions.  The first two were identical items, 28 Mhz blocks, and sold for 121 million and 134 million Swiss francs.  For the third auction, the item is a larger 56 MHz block. The price was only 55 million.  This is clearly far from equilibrium.  Not close to optimal welfare and low revenue as well.  Lesson learned: holding the single-item auction simultaneously, rather than sequentially.

20
Selling Items Separately  Rookie mistake 2: Use sealed-bid single-item auctions.  Imagine the items are identical and each bidder wants only one of them.  Two reasonable things to do: (1)pick one item and go for it (2)bid less aggressively on multiple items and hope toget one with a bargain price and not winning to many extra ones.  But which one to use? Tradeoff between winning too few and twinning too many.  The difficulty of bidding and coordinating gives low welfare and revenue sometimes.  Assume 3 bidders competing for two identical item, and each wants only one.

21
Selling Items Separately  In 1990, New Zealand government auctioned off essentially identical licenses for television broadcasting using simultaneous (sealed-bid) Vickrey auctions.  The revenue in the 1990 New Zealand auction was only $36 million, a paltry fraction of the projected $250 million.  On one license, the high bid was $100,000 while the second-highest bid (and selling price) was $6! On another, the high bid was $7 million and the second-highest was $5,000.  The high bids were made public... Every one can see how much money was left on the table...  They later switched to first-price auction, same problem remains, but at least less evident to the public...

22
Simultaneous Ascending Auctions (SAAs)  Over the last 20 years, simultaneous ascending auctions (SAAs) form the basis of most spectrum auctions.  Conceptually, it’s a bunch of single-item English auctions running in parallel in the same room.  Each round, each bidder place a new bid on any subset of items that she wants, subject to an activity rule.  Basically the rule says: the number of items you bid on should decrease over time as prices rise.

23
Simultaneous Ascending Auctions (SAAs)  Big advantage: price discovery.  This allows bidders to do mid-course correction.  Think about the three bidders two item case.  Another advantage: value discovery.  Finding out valuations might be expensive. Only need to determine the value on a need-to-know basis.